Characterizing Maximal Monotone Operators with Unique Representation

We study maximal monotone operators $A : X \rightrightarrows X^*$ whose Fitzpatrick family reduces to a singleton; such operators will be called uniquely representable. We show that every such operator is cyclically monotone (hence, $A=\partial f$ for some convex function $f$) if and only if it is 3-monotone. In Radon-Nikod\'{y}m spaces, under mild conditions (which become superfluous in finite dimensions), we prove that a subdifferential operator $A=\partial f$ is uniquely representable if and only if $f$ is the sum of a support and an indicator function of suitable convex sets.